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arXiv · 2601.04284

Sobre los teoremas de Shafarevich y Siegel

Hector Pasten
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Abstract

Presentaremos una nueva demostraci\'on del teorema de Shafarevich sobre finitud de curvas el\'ipticas con buena reducci\'on fuera de un conjunto finito de primos dado. Esto da un nuevo punto de entrada a teoremas fundamentales de finitud diofantina tales como el teorema de Siegel sobre la ecuaci\'on $S$-unidad. Nuestro argumento est\'a libre de aproximaci\'on diofantina o teor\'ia de trascendencia, y se acerca m\'as a las ideas de Faltings en su demostraci\'on de la conjetura de Mordell. -- We present a new proof of Shafarevich's theorem on finiteness of elliptic curves with good reduction outside a given finite set of primes. This gives a new entry point to fundamental diophantine finiteness theorems such as Siegel's theorem on the $S$-unit equation. Our proof is free from diophantine approximation or transcendence theory, and it is closer to the ideas of Faltings in his proof of Mordell's conjecture .

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Hector Pasten. 2026-01-07. Sobre los teoremas de Shafarevich y Siegel. https://arxiv.org/abs/2601.04284

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